Difference between revisions of "Subgroup"
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==Definition== | ==Definition== | ||
| − | Given a [[Group|group {{M|(G,\times_G:G\times G\rightarrow G)}} | + | Given a [[Group|group]] {{M|(G,\times_G:G\times G\rightarrow G)}} we say {{M|(H,\times_H:H\times H\rightarrow H)}} is a subgroup of {{M|(G,\times_G)}} if: |
| + | # <math>H\subset G</math> | ||
| + | # the function <math>\times_H:H\times H\rightarrow G</math> given by <math>\times_H(x,y)\mapsto\times_G(x,y)</math> has <math>\text{Range}(\times_H)\subseteq H</math> | ||
| + | #* That is to say it is closed. <math>\forall x\in H\forall y\in H[\times_H(x,y)\in H]</math> | ||
| + | # There exists an identity element <math>\in H</math>. | ||
| + | #* That is to say <math>\exists e\in H\forall x\in H[ex=xe=x]</math> where <math>xy</math> denotes <math>\times_H(x,y)</math> | ||
| + | # Every element has an inverse <math>\in H</math> | ||
| + | #* That is to say <math>\forall x\in H\exists y\in H[xy=yx=e]</math> | ||
| + | # The operation is associative | ||
| + | #* That is to say <math>\forall x\in H\forall y\in H\forall z\in H[x(yz)=(xy)z]</math> | ||
| + | |||
| + | Just like a [[Group|group]] | ||
| + | |||
| + | This makes it sound a lot harder than it really is. | ||
| + | |||
| + | ==Examples== | ||
| + | ===Even numbers=== | ||
| + | Take the group <math>(\mathbb{Z},+)</math> and define <math>H=\{z\in\mathbb{Z}|z\text{ is even}\}</math> then we have <math>H\subset\mathbb{Z}</math> and we must check it is a group. | ||
| + | |||
| + | # It is closed under <math>+</math> restricted to <math>H</math> - an even + an even = even. (proof <math>2n+2m=2(m+n)</math> and anything multiplied by 2 is even) | ||
| + | # The identity {{M|0\in H}} - so we have that. | ||
| + | # Given an <math>x\in H</math> we can see easily that the inverse, <math>-x</math> is also even and thus <math>\in H</math> | ||
| + | # Associativity is inherited | ||
| + | |||
| + | ==See also== | ||
| + | * [[Coset]] | ||
| + | * [[Normal subgroup]] | ||
| + | |||
{{Definition|Abstract Algebra}} | {{Definition|Abstract Algebra}} | ||
Latest revision as of 17:35, 15 March 2015
A subgroup [ilmath](H,\times_H:H\times H\rightarrow H[/ilmath] of a Group [ilmath](G,\times_G:G\times G\rightarrow G)[/ilmath] is a set [ilmath]H\subseteq G[/ilmath] which is a group under the operation [ilmath]\times_G[/ilmath] restricted to [ilmath]H\times H[/ilmath].
Contents
Definition
Given a group [ilmath](G,\times_G:G\times G\rightarrow G)[/ilmath] we say [ilmath](H,\times_H:H\times H\rightarrow H)[/ilmath] is a subgroup of [ilmath](G,\times_G)[/ilmath] if:
- [math]H\subset G[/math]
- the function [math]\times_H:H\times H\rightarrow G[/math] given by [math]\times_H(x,y)\mapsto\times_G(x,y)[/math] has [math]\text{Range}(\times_H)\subseteq H[/math]
- That is to say it is closed. [math]\forall x\in H\forall y\in H[\times_H(x,y)\in H][/math]
- There exists an identity element [math]\in H[/math].
- That is to say [math]\exists e\in H\forall x\in H[ex=xe=x][/math] where [math]xy[/math] denotes [math]\times_H(x,y)[/math]
- Every element has an inverse [math]\in H[/math]
- That is to say [math]\forall x\in H\exists y\in H[xy=yx=e][/math]
- The operation is associative
- That is to say [math]\forall x\in H\forall y\in H\forall z\in H[x(yz)=(xy)z][/math]
Just like a group
This makes it sound a lot harder than it really is.
Examples
Even numbers
Take the group [math](\mathbb{Z},+)[/math] and define [math]H=\{z\in\mathbb{Z}|z\text{ is even}\}[/math] then we have [math]H\subset\mathbb{Z}[/math] and we must check it is a group.
- It is closed under [math]+[/math] restricted to [math]H[/math] - an even + an even = even. (proof [math]2n+2m=2(m+n)[/math] and anything multiplied by 2 is even)
- The identity [ilmath]0\in H[/ilmath] - so we have that.
- Given an [math]x\in H[/math] we can see easily that the inverse, [math]-x[/math] is also even and thus [math]\in H[/math]
- Associativity is inherited
